Additionally, we have used the Q-criterion to present a three-dimensional visualisation of the vortices in the present study. The Q-criterion was described by [1]} as a quantity which measures the dominance of vorticity \(\omega \)  over strain \(s\) ,
\(Q=1/2 ( \left\Vert  \omega \right\Vert ^2 - \left\Vert  s \right\Vert ^2 )\) . Figure REF  shows the spatial development of the coherent structures for \(Q=0.1\) . Near the injection point, axisymmetric KH vortex rings are located close to the surface (a magnified view is shown). As explained previously, when \(\omega _{x}\)  and \(\omega _{\theta }\)  are of comparable magnitude, the KH vortices are stretched downstream or upstream.
The topological shape of these vortical structures resembles the instantaneous hairpin-like vortical structures reported in experiments and numerical simulations by [2]}, [3]}, [4]}. Outer hairpin vortices (a magnified view is also shown) are observed clearly and the inner hairpin vortices not as clearly as they are localised underneath the interface. Further downstream, we observe the presence of a `cap-vortex' covering the mushroom-like structure. Inside of this structure, the vortices are unstable and break down.
